Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 218

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\[1)\ 7^{x} - 7^{x - 1} = 6\]

\[7^{x} \bullet \left( 1 - 7^{- 1} \right) = 6\]

\[7^{x} \bullet \left( \frac{7}{7} - \frac{1}{7} \right) = 6\]

\[7^{x} \bullet \frac{6}{7} = 6\]

\[7^{x} = 7\]

\[x = 1\]

\[Ответ:\ \ x = 1.\]

\[2)\ 3^{2y - 1} + 3^{2y - 2} - 3^{2y - 4} = 315\]

\[3^{2y} \bullet \left( 3^{- 1} + 3^{- 2} - 3^{- 4} \right) = 315\]

\[3^{2y} \bullet \left( \frac{1}{3} + \frac{1}{9} - \frac{1}{81} \right) = 315\]

\[3^{2y} \bullet \left( \frac{27}{81} + \frac{9}{81} - \frac{1}{81} \right) = 315\]

\[3^{2y} \bullet \frac{35}{81} = 315\]

\[3^{2y} = 729\]

\[3^{2y} = 3^{6}\]

\[2y = 6,\]

\[y = 3\]

\[Ответ:\ \ y = 3.\]

\[3)\ 5^{3x} + 3 \bullet 5^{3x - 2} = 140\]

\[5^{3x} \bullet \left( 1 + 3 \bullet 5^{- 2} \right) = 140\]

\[5^{3x} \bullet \left( \frac{25}{25} + \frac{3}{25} \right) = 140\]

\[5^{3x} \bullet \frac{28}{25} = 140\]

\[5^{3x} = 125\]

\[5^{3x} = 5^{3}\]

\[3x = 3\]

\[x = 1\]

\[Ответ:\ \ x = 1.\]

\[4)\ 2^{x + 1} + 3 \bullet 2^{x - 1} - 5 \bullet 2^{x} + 6 =\]

\[= 0\]

\[2^{x} \bullet \left( 2^{1} + 3 \bullet 2^{- 1} - 5 \right) = - 6\]

\[2^{x} \bullet \left( \frac{3}{2} - 3 \right) = - 6\]

\[2^{x} \bullet \left( \frac{3}{2} - \frac{6}{2} \right) = - 6\]

\[2^{x} \bullet \left( - \frac{3}{2} \right) = - 6\]

\[2^{x} = 4\]

\[2^{x} = 2^{2}\ \]

\[x = 2\]

\[Ответ:\ \ x = 2.\]